Determination of the singularities of the Dyson-Schwinger equation for the quark propagator

In this paper we study the analytic continuation of the Dyson-Schwinger equation into the entire complex p2 plane. It is shown that in the usual ladder approximation with a running coupling, there are two complex-conjugate branch points and one real branch point in the timelike region. The real branch point corresponds presumably to a physical mass for a quark; but it is removed by the addition of a so-called confinement term to the bare gluon propagator: with this extra term, there are at least three pairs of complex-conjugate branch points. The origin of these complex singularities is not yet clear.